Remarks on W^{1, p}-quasiconvexity, interpenetration of matter, and function spaces for elasticity

We show that, under mild hypotheses on the elastic energy function, the minimizer of the energy in the space {f∈W^1,p:det∇f>0,1≦p<n} of a nonlinear elastic ball subject to the severe compressive boundary conditions f(x)=λx,λ≪1 will not be the expected uniform compression f (x) = λx. To show this, we construct competitors in this space that reduce the energy but interpenetrate matter. We also prove that the W^{1, p}-quasiconvexity condition of Ball and Murat [1984] is a necessary condition for a local minimum in a setting that includes nonlinear elasticity. This theorem is well suited to analyses of the formation of voids in nonlinear elastic materials. Our analysis illustrates the delicacy of the choice of function space for nonlinear elasticity.